Klein 2-Geometry VII

November 1, 2006

Posted by David Corfield

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Let’s reconvene the latest session of the Honorable Guild of Categorifiers of Kleinian Geometry. I’ll briefly sum up what I learned from last month’s efforts. Our plan had been to work out the projective 2-space associated to a Baez-Crans (BC) 2-vector space, find the 2-group of projective linear transformations, and then study sub-2-groups, in order to throw up 2-geometries which were categorifications of sub-geometries of projective geometry such as Euclidean or spherical geometry. But Urs posed for us the task of finding the general linear 2-group of such 2-vector spaces, and he helped it see the light of day, we think, here.

This suggested another path to Euclidean 2-geometry if we could find a way to put an inner product on a BC 2-vector space, and then look at the sub-2-group of transformations which preserve it. However, we met with a small problem. I wondered whether we might look to other forms of 2-vector space, $C$ -modules for categories other than Disc( $k$ ), such as what we called (1,1) vector 2-spaces.

Elsewhere, David Roberts wondered whether we could use 2-ordinals to keep track of incidence relations between the objects of our 2-geometries.

Tim Silverman joined the team and wrote many comments, perhaps he would like to sum up his discoveries.

Posted at November 1, 2006 9:19 AM UTC

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categorified scalar product

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put an inner product on a BC 2-vector space

I have changed my mind about the nicest way to put an inner product on the 2-space. It might be good to use categories internal to hermitean vector spaces instead of just plain vector spaces. Then a good notion of categorified inner product should be given by the internal $hom$ , making contact with the way I envisioned generalized Hilbert spaces of states here.

Posted by: urs on November 1, 2006 10:08 AM | Permalink | Reply to this

Re: categorified scalar product

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categories internal to hermitean vector spaces

Can these be seen as $C$ -modules for some category $C$ ?

Posted by: David Corfield on November 1, 2006 3:32 PM | Permalink | Reply to this

Re: categorified scalar product

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categories internal to hermitean vector spaces

Can these be seen as C-modules for some category C?

I would say they are still $Disc (k)$ -modules - but with extra structure.

Like an ordinary hermitean $k$ -vector space is still a $k$ -module, but with additional structure.

Posted by: urs on November 1, 2006 6:08 PM | Permalink | Reply to this

Re: categorified scalar product

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So what’s the sub-2-group of the 2-group you found (the Schreiber 2-group?) of linear transformations of a BC 2-vector space which preserve this additional structure? Something like, when $k = ℂ$ , $G = U (n) \times U (m)$ worth of objects, with some subgroup of linear maps from $k^{n}$ to $k^{m}$ as morphisms between two objects, and the restricted action of the one you described?

Posted by: David Corfield on November 2, 2006 12:34 PM | Permalink | Reply to this

Re: categorified scalar product

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[…] preserve this additional structure?

I am afraid, unfortunately, that I still haven’t said exactly what additional structure I want to regard as a scalar product on a BC 2-vector space.

But, as you indicate, once we agree on one definition, we should certainly compute the sub2-group of the general linear 2-transformation preserving this.

Posted by: urs on November 2, 2006 1:54 PM | Permalink | Reply to this

Re: categorified scalar product

According to your characterisation as categories internal to the category of hermitian vector spaces, I take it that the inner product on arrows applies to the whole vector space of arrows. If we stick with skeletal such 2-vector spaces, would you envisage there being non-zero inner products between arrows based at different objects?

Posted by: David Corfield on November 2, 2006 3:12 PM | Permalink | Reply to this

Re: categorified scalar product

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According to your characterisation […]

The trouble is, I keep changing my mind about what a good notion of inner product would be. Sorry for that.

Right now I am trying to find one that generalizes the notion of the Killing form on a Lie algebra.

So consider for a moment a BC 2-vector space with the structure of a Lie 2-algebra on it. Call it

(1)

L .

We get a representation of $L$ on itself: the adjoint 2-rep

(2)

ad : L \to End (L) .

Bracketing by identity morphisms on objects $x \in Obj (L)$ yields linear functors $L \overset{[x, \cdot]}{\to} L$ , and morphisms $(x \overset{f}{\to} y) \in Mor (L)$ yield linear natural transformation between these

(3)

\begin{matrix} ↗ ↘^{ad (x)} \\ L & ⇓ ad (f) & L \\ ↘ ↗_{ad (y)} \end{matrix} .

The point is that it might be easier to see the right inner product on $ad (L)$ , because that should come from a categorification of the trace.

For monoidal 1-categories with duals on objects, there is a nice way to write the trace of any morphism by closing the string representing the morphism on itself.

Something similar should hold here, but right now I have no good grasp of it.

But as a warmup, we can compute “single matrix elements”, at least:

Consider $Disc (k)$ as a BC 2-vector space. We have

(4)

L ≃ Hom (Disc (k), L) .

So yet another way to think of a morphism in $L$ is as a natural transformation

(5)

\begin{matrix} ↗ ↘^{x} \\ Disc (k) & ⇓ f & L \\ ↘ ↗_{y} \end{matrix} .

Similarly, a co-2-vector would be

(6)

\begin{matrix} ↗ ↘^{x^{*}} \\ L & ⇓ f^{*} & Disc (k) \\ ↘ ↗_{y^{*}} \end{matrix} .

If we let ${x_{i} \overset{f_{j}}{\to} x_{i} + t (f_{j})}$ be a basis for $L$ , and denote the dual basis by stars, then the $(i, k)$ -object-entry of the matrix representing

(7)

ad (x \overset{f}{\to} y)

in that basis would be

(8)

Disc (k) \overset{x_{i}}{\to} L \overset{ad (x)}{\to} L \overset{x_{k}^{*}}{\to} Disc (k) .

Same as for 1-vector spaces.

But now we possibly also want to look at morphisms from and into $Σ (k)$ to get matrix elements at the level of morphisms

(9)

\begin{matrix} ↗ ↘ & ↗ ↘^{ad x} & ↗ ↘ \\ Σ (k) & ⇓ r & L & ⇓ ad f & L & ⇓ r^{*} & Σ (k) \\ ↘ ↗ & ↘ ↗_{ad y} & ↘ ↗ \end{matrix} .

Maybe now we can sum over $r$ to get a trace and hence a scalar product.

But I don’t know. Have to run now.

Posted by: urs on November 3, 2006 4:20 PM | Permalink | Reply to this

equivalence classes of n–vector spaces

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I have asked the experts and found the following answer to a crucial question in this BC $n$ -vector business:

Question: We may regard an $n$ -term chain complex as living in an $n$ -category, whose morphisms are chain maps, 2-morphisms are chain homotopies, and so on.

Are equivalence classes in this $n$ -category in bijection with cohomology classes, i.e. with chain complexes all whose maps are the 0-map?

Answer: yes, this is true. It is in fact true for chain complexes in any category of projective modules.

This means that whenever one sees people inverting “quasi isomorphisms” in the 1-category of chain complexes, this is a 1-categorical workaround for talking about equivalence in a setup that is really $ω$ -categorical.

Hence forming the derived category of chain complexes is a lot like working with the full $ω$ -category structure on chain complexes in the first place (i.e. with reagrding arbitrary chain complexes as $ω$ -vector spaces).

I gather that this is very well known to the experts. But I didn’t know it before.

Posted by: urs on November 1, 2006 10:54 AM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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David said, looking pointedly at me,

Tim Silverman joined the team and wrote many comments, perhaps he would like to sum up his discoveries.

Er … [stands nervously at front of class]

[fiddles with papers]

[squeaky voice … ]

I think that blizzard of comments was more a sign of me getting myself oriented, spotting connections between things, and going a short way up a lot of paths, than producing technical results of any interest. Given that, I think I’ll summarise my overall impression of the directions we might go in, rather than saying too much about the ground I’ve covered so far.

One danger that we seem to have encountered is that, if we start from a beautifully symmetrical and simple Klein 1-geometry and try to categorify it, the result may be rather boring, in the sense that it won’t really introduce anything new. I think this showed up with the 2-vector spaces. It appears to me that JB and David all but cleared this out right back at the beginning of that discussion, and most of what was left was pretty much finished off by Urs.

Of course, my opinion about this may perhaps just be due to my own idiosyncratic view of what counts as boring and what as interesting …

In the course of trying to get round this, I explored a couple of different paths:

1) Looking at small, runty symmetry 2-groups instead of large, beautiful ones, or examining small, runty sub-2-groups of large beautiful 2-groups. I think (in retrospect) that this is more-or-less what I was doing in the series of posts where I looked at the 2-term chain complexes of 0-forms and 1-forms over various graphs. I think that the symmetries of the graphs induce unexpected and interesting sub-2-groups of the automorphism 2-group of the chain complex itself, along with some at least mildly interesting quotient spaces. I got a bit confused and bogged down with this, from posting too fast and thinking too slowly, but I think (or hope) there is still a lot of mileage in this approach and I want to come back to it when I have sorted out some of the sub-issues in my mind.

2) Avoiding going straight down to the skeletal 2-space. I seem to be a bit out of step with everybody else here. Of course, I don’t want to be evil (i.e. forget equivalences and only think about isomorphisms). At least, not very evil. On the other hand, I still think the process by which equivalences are carried out has the potential to be interesting in particular cases. It’s not like the difference between $R^{n}$ and a single point is of no interest whatever …

From the particularities of 2-vector spaces, I then jumped up to the general case of beautifully symmetrical 2-spaces, in particular taking 1-spaces with a transitive group action on them, and constructing internal categories in categories of these spaces. While this symmetry requirement to some extent discards the interest of going to 2-spaces (e.g. it avoids going in the orbifold/stack direction of keeping track of different kinds of points), it at least lets us get an overview of all those boringly symmetrical 2-spaces.

This ‘internal category’ approach more-or-less forces us to think about the 2-space as a groupoid, and to think about the relationship between this groupoid, and the weak quotient groupoid of the object 1-space by its automorphism group (qua 1-space).

The groupoid view places at least two phenomena in front of us: isomorphisms between different points, and automorphisms of single points. It is probably a good idea to look at these separately, at least initially, although they have a good deal in common.

I tried approaching both of these through a fibre-bundle view. We can, perhaps, think of the automorphisms of a point as forming a fibre over that point; and of the objects and morphisms in one component of a groupoid as forming a kind of categorified fibre over the component (i.e. thinking of a base space whose points are the components, with the fibres being the internal structure of the components). The most obvious odd thing (to me) about this manner of thinking is that the paths in the space are global automorphisms of the base space, rather than, e.g. smooth curves. This is of course usual for Klein geometry.

I think this bundle approach, and related approaches, are worth exploring in more depth, because they may quickly connect up to a lot of other things. Judging by other threads, they’re also, perhaps, the approaches most likely to be of interest to other participants here. But I’m kind of reluctant to do too much on this, because all those other participants know such an awful lot more about this sort of thing than I do … so anything I say is liable to sound rather babyish.

At the moment, I’m having a go at working on the problem from the opposite direction: instead of picking a 2-space and trying to understand its 2-symmetry group, its figures, and its figure-symmetry sub-2-groups, I’m trying to pick a particular, very simple, 2-group and look at its various actions, as well as generally trying to understand the structure of the 2-group as thoroughly as possible.

There seem to be a very large number of different generalisations of the concepts of ‘action’ and ‘quotient’. I want to understand these and classify them. Once I’ve done that, I think a bunch of other things will become clearer.

One other point I’d like to make is that ordinary group theory is a huge, complicated subject in its own right. It’s not surprising if the theory of 2-groups has a lot of stuff lurking below the surface which isn’t simply a straightforward generalisation of the 1-group case. There’s really a tremendous lot to discover here. I really ought to familiarise myself with the existing literature, but I’m not sure I’m even at the point where I can understand it, except maybe at a very superficial and useless level.

I very much like Dave Roberts’ idea of attacking generalisations of the subobject lattice, and looking at 2-dimension via 2-ordinals. On the other hand, I’m increasingly of the opinion that the best way to categorify incidence geometry is to creep up on it from behind, rather than coming at it head-on.

To summarise the summary:

2-symmetry 2-groups which are categorifications of symmetry groups seem to have interesting 2-subgroups which are not simply categorifications of the subgroups of the symmetry groups.

Not all spaces are skeletal, and equivalences have interest beyond their mere existence!

There are some interesting things to be said about spaces which are groupoids in two different ways.

It may be interesting to think about groupoids as fibre or 2-fibre bundles.

It may be worth picking a 2-group and understanding it in detail, looking at the ways it can act as a 2-symmetry 2-group. This should involve understanding its sub-2-group lattice, and getting a better handle on what a quotient is.

Generally, we can expect interesting 2-groups to contain a lot of stuff that goes beyond what we can understand in terms of 1-groups.

To summarise the summary of the summary

Graphs, groupoids, bundles, quotients … ohhh, it’s all hard!

Posted by: Tim Silverman on November 2, 2006 8:19 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

Urs has a nice clear formulation here of the action of morphisms on 2-morphisms in the general linear 2-group of transformations on a BC 2-vector space. Perhaps we might consider possible sub-2-groups of this 2-group. Presumably we can form those which are full on 2-morphisms but which restrict independently on the two components of the 1-morphisms, e.g., to orthogonal groups.

On the other hand, if we go for non-full on 2-morphisms options, then we’ll have to be careful that the action of the 1-morphisms preserves them.

Posted by: David Corfield on November 8, 2006 2:20 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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Back here:

Might it have appeared in HDA5?

I don’t recall having seen it there. What does appear is the slightly similar Poincaré 2-group.

Urs commented on the similarity between his 2-group and the Poincaré 2-group.

I guess actually it’s just a full sub-2-group of his 2-group for $p = 4$ and $q = 1$ , with the product of the trivial subgroup of GL(1) and the Lorentz subgroup of GL(4) for 1-morphisms.

This perspective opens up a huge range of 2-groups. We might also look at some arithmetic ones, with 1-morphisms some subgroup of $SL (n, Z) \times SL (m, Z)$ , and 2-morphisms maps between $Z^{n}$ and $Z^{m}$ . When do we start on the 2-Langlands Program?

Then we might try permutation groups, although perhaps these are just going to be sub-2-groups of Urs’ 2-group for the ‘field with one element’ (see here, TWF184 and TWF187, and Tuesday 8 entry here).

Posted by: David Corfield on November 9, 2006 9:02 AM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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I guess actually it’s just a full sub-2-group of his 2-group for $p = 4$ and $q = 1$ […]

Oh, good point! Yes, that seems to be right. Interesting.

(The group of morphisms, strictly speaking, is $Hom (k^{4}, k^{1})$ , hence covectors of $k^{4}$ instead of vectors, but of course that’s completely immaterial.)

When do we start on the 2-Langlands Program?

Uh, er - as soon as I find the time I’ll join you on that project. Meanwhile we have to stick to Kapranov’s observation on categorified Langlands.

Posted by: urs on November 9, 2006 9:48 AM | Permalink | Reply to this

Re: Klein 2-Geometry VII

I see (ref 23) John’s been working on the representation theory of the Poincaré 2-group. Would our discovery that it’s a sub-2-group help? Perhaps there’s a 2-adjunction between 2-categories of 2-representations, corresponding to the adjunction between induced and restricted representations at the level below.

Posted by: David Corfield on November 9, 2006 1:13 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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representation theory of the Poincaré 2-group

I am looking forward to seeing that.

I am aware of the Crane-Sheppeard work on reps of the Poincaré 2-group #.

It seems that in this context people always only consider 2-reps on Kapranov-Voevodsky 2-vector spaces (or a certain infinite-dimensional generalizations of them).

As I apparently don’t get tired of mentioning, it strikes me that we should really be looking, much more generally, at reps in bimodules. That’s because we have the chain of inclusions

(1)

KV 2 Vect \overset{\subset}{\to} Bim \overset{\subset}{\to}_{Vect} Mod

and because reps in $Bim$ show up at various places in string physics. These reps yield associated bundle gerbes, associated nonabelian bundle gerbes and string-bundles with connection, for instance.

And what is nice, for every strict 2-group

(2)

G_{2} = (H \to G)

we get a canonical 2-rep

(3)

ρ : Σ (G_{2}) \to Bim

which works like

(4)

ρ : \begin{matrix} ↗ ↘^{g} \\ • & h ⇓ & • \\ ↘ ↗^{g'} \end{matrix} \mapsto \begin{matrix} ↗ ↘^{ℂ [H]_{g}} \\ ℂ [H] & \cdot h ⇓ & ℂ [H] \\ ↘ ↗^{ℂ [H]_{g'}} \end{matrix},

where $ℂ [H]$ is the group algebra of $H$ , and $ℂ [H]_{g}$ is this algebra regarded as a bimodule over itself, with the right action twisted by $α (g) \in Aut (H)$ .

I have no proof, but several indications that we should really be looking at 2-reps in bimodules this way. Apart from the fact that I know a couple of places where these actually appear prominently in applications, the reason is that

a) $Bim$ is much larger than $KV 2 Vect$ . And for instance the work by Baas/Dundas/Rognes/Richter indicates that 2-vector bundles with fibers in $KV 2 Vect$ are not general enough, while the work by Stolz/Teichner indicates that those with fibers in $Bim$ are.

b) several people indicated to me that the inclusion

(5)

Bim \overset{\subset}{\to}_{Vect} Mod

should in fact be an equivalence

(6)

Bim ≃_{Vect} Mod

if conditions are chosen suitably.

So this then would mean that 2-reps in $Bim$ capture actually all possible $Vect$ -linear 2-reps.

But I am still looking for an actual proven statement of this supposed equivalence. If anyone out there knows about this, please drop me a note.

Would our discovery that it’s a sub-2-group help?

Hm, maybe. I’d need to think about that.

Posted by: urs on November 9, 2006 2:31 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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For some strange reason I wrote:

I am still looking for an actual proven statement

Actually, I do know the answer.

Posted by: urs on November 9, 2006 8:56 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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I wrote:

Actually, I do know the answer.

So let me get this straight, finally.

Viktor Ostrik’s theorem 1 (p.10) says that for $C$ a sufficiently well-behaved abelian monoidal category, the canonical monomorphism

(1)

Bim (C) \to_{C} Mod

is essentially surjective on indecomposable objects, i.e. surjective on equivalence classes of indecomposable objects.

Here “sufficiently well behaved” means $C$ is semisimple, rigid, has finitely many irreducible objects (but see remark 9) and has irreducible tensor unit.

A module category is indecomposable if it is not the direct sum of two nontrivial module categories.

Examples of categories $C$ with these properties are given on p. 5. Finite dimensional vector spaces are among them.

So:

we have a sequence of monomorphisms

(2)

KV 2 Vect \overset{\subset}{\to} Bim (Vect) \overset{\subset}{\to}_{Vect} Mod

with the last one being essentially surjective on indecomposable objects.

Posted by: urs on November 10, 2006 5:19 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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Does the reason for the physical interest in the Poincaré 2-group relate to the interest in the Poincaré group? If so, wouldn’t the double cover of the 2-group be worth looking at?

I wonder how general we can make our family of skeletal 2-groups. Presumably any pair of groups $G$ and $H$ with a left and right action, respectively, on an abelian group of 2-morphisms, $K$ would do. Is there a term for what is to a representation as a bimodule is to a module? Birepresentation?

Posted by: David Corfield on November 10, 2006 3:00 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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Would it be useful to combine an internal symmetry group with the Poincaré 2-group? Something like 1-morphisms are $G \times H$ , where $G$ is $SO (3,1)$ (or its double cover) and $H$ is $SU (3) \times SU (2) \times U (1)$ (divided by $Z_{6}$ if you like). Then for 2-morphisms one would need at least $R^{4}$ , and one could keep with that allowing $H$ to act trivially. On the other hand, there is scope to allow a larger space of 2-morphisms and a nontrivial action by $H$ .

Posted by: David Corfield on November 11, 2006 10:44 AM | Permalink | Reply to this

Re: Klein 2-Geometry VII

For what it’s worth, a paper on representations of the Poincaré 2-group and state sum models should appear in a month or so, by Laurent Freidel, Aristide Baratin, and some other authors possibly including myself. Here the Poincaré 2-group is represented on 2-Hilbert spaces of the Crane-Yetter sort, following the ideas of Crane and Sheppeard.

Posted by: John Baez on November 13, 2006 7:39 AM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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There must be a name for what I’m calling a birepresentation. It’s like a profunctor except taking values in Vect.

$G \times H^{op} \to Vect$

I was considering groups $G$ and $H$ (as one object categories), but any categories would do. Either way, could we then compose these bireps using Kan extensions as is done for profunctors here? If that’s right that every birep of groups gives a (strict) 2-group, then we’d have a new type of composition.

Posted by: David Corfield on November 13, 2006 9:04 AM | Permalink | Reply to this

Re: Klein 2-Geometry VII

Probably a trivial exercise with enriched coends, if I knew anything about them. For a similar use, but over a different category, see pp. 13 -14 of Lawvere’s Taking Categories Seriously.

I like the “Rejecting the complacent description of that [aforementioned] identification as a mere analogy or amusement, its relentless pursuit is continued” in the abstract. Here’s to the relentless pursuit of category theoretic ideas into concrete situations - Klein 2-geometry is no mere analogy or amusement either.

Posted by: David Corfield on November 13, 2006 6:13 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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David observed that the basic structure of what we seem to have identified as the general linear 2-group (on a skeletal 2-vector space) apparently generalizes to a situation where

the group of objects is a product $G_{1} \times G_{2}$

the group of morphisms is the additve group of a linear vector space $V$

the target map is trivial

objects act on morphisms by means of a birep

$G_{1}^{op} \times G_{2} \to Vect$ .

Inspired by this he observes:

Either way, could we then compose these bireps using Kan extensions as is done for profunctors here? If that’s right that every birep of groups gives a (strict) 2-group, then we’d have a new type of composition.

I see how the appearance of bireps (or profunctors, if one wishes) anywhere makes one want to define their composition - which is only natural.

But for the present context I seem to be at a loss with respect to understanding what this composition would mean abstractly for a 2-group.

And in fact, it seems that only on the morphisms a natural composition is induced. Do we really get a map

first 2-group times second 2-group to third 2-group

this way, that respects any of the structure around?

Posted by: urs on November 13, 2006 6:44 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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So you’re agreeing that the coend is possible, but don’t know whether it has a meaning? Or is there something about Vect that doesn’t allow enriched coends? Mimicking the quotienting process for Set, one would expect something like $W \times V / ~$ where $(w', v') ~ (w, v)$ if there’s an element of the middle group which acts on $w$ on the left to give $w'$ and on $v'$ on the right to give $v$ . Oh, where $V$ and $W$ are the vector spaces involved in the bireps.

I’ll see if it has a meaning.

Posted by: David Corfield on November 13, 2006 7:14 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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So you’re agreeing that the coend is possible, but don’t know whether it has a meaning?

Yes. I haven’t thought about it, but I would expect the composition of bireps you have in mind exists.

But I am puzzled by what that should mean for the corresponding 2-groups.

Posted by: urs on November 13, 2006 7:38 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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No meaning yet, but I kinda convinced myself that if we’re composing $(G, H)$ with birep $V$ and $(H, K)$ with birep $W$ , then the action of $(G, K)$ on that quotient I mentioned is the sum over h of $({hwk}^{- 1}, {gvh}^{- 1})$ . Seems to be invariant over the quotient class.

I’m casually slinging round the term ‘birep’, but does it exist? Maybe, there’s no point to it as a group is iso to its opposite and so a birep is just the same as a rep of the product. Unless, of course, we can compose suitable bireps, like bimodules.

Posted by: David Corfield on November 13, 2006 7:45 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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I’ll just say some things I already said in my recent long post, which may have been buried amid technicalities.

David wrote:

I’m casually slinging round the term ‘birep’, but does it exist?

I’ve never heard of it, so if you want to sound like everyone else, you can say “bimodule” instead.

Maybe, there’s no point to it as a group is iso to its opposite and so a birep is just the same as a rep of the product.

Right, there’s not much need to talk about birepresentations, since they’re just representations of a product of groups.

But, for the same reason, there isn’t much need to talk about bimodules of rings or algebras, and people do about those.

The advantage of having special words like this is that bimodules and birepresentations can be thought of as morphisms going from one ring, or algebra, or group, to another.

Unless, of course, we can compose suitable bireps, like bimodules.

You can indeed, since a birepresentation is a bimodule!

A representation of a group $G$ is the same as a module of its group algebra $k [G]$ , where $k$ is your favorite field.

A $G, H$ -birepresentation is thus the same as a $k [G], k [H]$ -bimodule, or in other words a module of $k [G] \otimes k [H]$ , or in other words a module of $k [G \times H]$ .

So, composing birepresentations is just a special case of composing bimodules! If you hadn’t already defined how to compose birepresentations, this is how you should define it.

All this stuff is very, very related to Hecke algebras and Hecke operators - as I’ll explain someday, when I get around to saying what Jim has done over the last 2 years.

Posted by: John Baez on November 13, 2006 10:54 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

All this stuff is very, very related to Hecke algebras and Hecke operators - as I’ll explain someday, when I get around to saying what Jim has done over the last 2 years.

That’s reassuring. You’ve mentioned that work before. Do you realise the anxiety you cause by continually making additions to the list of things that Jim has worked on which may never see the light of day?

Posted by: David Corfield on November 14, 2006 8:48 AM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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The earlier mention was from Klein 2-geometry II:

In particular, the subject of $G$ -invariant binary relations underlies the theory of Hecke operators and Hecke algebras. This is a little appreciated fact, because most people working on Hecke algebras focus on very special groups $G$ - namely, finite reflection groups or $SL (2, Z)$ . Jim has been working out the general theory, and someday I’ll have to explain a bit of it on This Week’s Finds.

Perhaps I strayed close with my suggested 2-group here - 1-morphisms some subgroup of $SL (n, Z) \times SL (m, Z)$ , and 2-morphisms maps between $Z^{n}$ and $Z^{m}$ .

Posted by: David Corfield on November 14, 2006 10:03 AM | Permalink | Reply to this

Re: Klein 2-Geometry VII

Do you realise the anxiety you cause by continually making additions to the list of things that Jim has worked on which may never see the light of day?

Fortunately, Jim gives talks about this stuff at UCR sometimes. So there is a fair chance that any particular idea will be explained in somebody’s notes. Nevertheless, it’s clear to me that his notebooks will keep researchers (after his death) busy for many years, full of unpublished but important and prescient ideas, like those that we find now in the notebooks of Leibniz. (OK, Jim probably won’t be as significant as Leibniz.)

Posted by: Toby Bartels on November 16, 2006 6:30 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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Back to the question of whether composition of bireps has any meaning for the corresponding 2-groups. Let’s consider the full general linear 2-groups for BC 2-vector spaces $k^{p, q}$ and $k^{q, r}$ . The former has as 2-morphisms functions between $k^{p}$ and $k^{q}$ , and similarly for the latter. Now the tensor product is formed of pairs of 2-morphisms $(f, g)$ , but where $(f . a, g)$ is identified with $(f, a . g)$ for $a$ in $GL (q)$ . So these 2-morphisms are maps from $k^{p}$ to $k^{r}$ which factor through $k^{q}$ , and we don’t care how they factor.

Don’t you ever do something like that with bundles, where a fibre for one bundle is the base for another bundle?

Posted by: David Corfield on November 14, 2006 9:27 AM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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I should have said:

Don’t you ever do something like that with bundles, where a total space for one bundle is the base space for another bundle?

Hmm, so is there a category of topological spaces with maps continuous functions, $f : X \to Y$ such that $f$ is a vector bundle, and the obvious composition. Then we’re looking at how the corresponding 2-groups of gauge transformations compose.

Posted by: David Corfield on November 14, 2006 11:32 AM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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Don’t you ever do something like that with bundles, where a total space for one bundle is the base space for another bundle?

Yes, we do. But could you explain again in which sense this is related to the composition of 2-groups that you have in mind? I am not sure yet that I see what you mean.

Posted by: urs on November 16, 2006 8:01 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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The intuition, which may of course be wrong, is: We can compose two BC 2-vector spaces, $k^{p, q}$ and $k^{q, r}$ to form a $k^{p, r}$ space, noting the composition is skeletal. Then the composition of 2-groups I’m describing corresponds to the composition of the general linear 2-groups associated to the 2-vector spaces.

Posted by: David Corfield on November 17, 2006 9:09 AM | Permalink | Reply to this

Re: Klein 2-Geometry VII

I had been thinking to myself that this composition of 2-groups feels like composition of species (aka structure types), see here. And of course the latter is also just another example of a coend construction. There’s a paper by Fiore, Gambino, Hyland and Winksel which treats generalised species. Our ordinary ones are (1,1) species, a certain kind of profunctor. Composing species is then a form of composing profunctors (a bit more subtle than the usual composition, see p.10 onwards).

On another point, the authors say:

More recently, Baez and Dolan considered further generalisations of these structures leading to the concepts of sorted symmetric set-operad [1] and of stuff types [2] (see also [39]). The former, though not the latter, can be directly recast in our setting (p. 2)

So, I wonder how stuff types and composition of stuff types (p. 31) fits into the picture.

Then where is the theory of generalised stuff types?

If an operad is a monoid object in the category of structure types with composition as the monoidal structure, how about in the category of generalized structure types (species)? And in the category of stuff types?

Elsewhere John spoke of the search for gold nuggets. Sometimes they seem to be so liberally sprinkled around in the rock that it’s hard to know which to pick out first. Admittedly picking them out may not be as simple as it looks.

Posted by: David Corfield on November 17, 2006 4:58 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

David writes:

[…] is there something about Vect that doesn’t allow enriched coends?

No, it’s a cocomplete symmetric monoidal category so you’re fine.

(I’m assuming that you’re allowing infinite-dimensional vector spaces here.)

Posted by: John Baez on November 13, 2006 9:55 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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Okay, I’ll do a little work today. David writes:

There must be a name for what I’m calling a birepresentation. It’s like a profunctor except taking values in Vect. $G \times H^{op} \to Vect$

I assume $G$ and $H$ here are groups, or maybe groupoids, or maybe categories?

If they’re groups, ordinary mortals would call this thing a “representation of $G \times H$ ”, since $H^{op} ≅ H$ for groups.

If they’re categories, I might call this thing a “ $Vect$ -enriched profunctor”. Australians would call it a “bimodule” or simply a “module”.

That last fact deserves some explanation. This spring in Chicago, Steve Lack gave a bunch of talks on Australian $n$ -category theory. The notes will eventually show up in the proceedings of the IMA conference. He explained that these days, Aussies are very fond of doing everything with profunctors instead of functors. And, ever since Kelly, they’ve been fond of doing everything in an enriched context whenever possible. So, anything you or I can do with a functor $F : C \to D$ they can do with a $K$ -enriched profunctor $F : C^{op} \times D \to K$ where $K$ is a sufficiently nice category, like $Vect$ .

This conceptual move is a lot like replacing functions from $S$ to $T$ by $S \times T$ -shaped matrices - going from set theory to linear algebra. Any function gives a matrix of 0’s and 1’s, so one doesn’t lose much - and one gains a lot.

But, they get sick of saying “ $K$ -enriched profunctor”, so they say “bimodule” or simply “module”.

Note that in ordinary ring theory, an $A, B$ -bimodule is just a module over $A^{op} \otimes B$ . So, there’s no real need for the word “bimodule” in that context, if one is feeling hasty. The same thing is happening here! In fact, if one is clever enough, one can see that what we’re doing subsumes the ordinary theory of bimodules of rings. (There’s a little wrinkle here which you’ll see if you try to actually to do this, but it’s no big deal.)

Anyway, in ordinary ring theory we get a bicategory of

rings
bimodules
bimodule homomorphisms

So, here we get a bicategory of

categories
$K$ -enriched profunctors
natural transformations between $K$ -enriched profunctors

and everything works more or less the same if $K$ is sufficiently nice - which it is in your example.

I don’t think you need to formally know anything about “ $K$ -enriched coends” to deal with this stuff at an intuitive level. When you compose $K$ -enriched profunctors, it’s just a categorified version of matrix multiplication! The formula for matrix multiplication involves a sum of products. Similarly, the formula for composing $K$ -enriched profunctors involves a “weighted colimit of tensor products”. This sort of thing is also called a “ $K$ -enriched coend”, but don’t let that perturb you. The main thing is, it works whenever $K$ is sufficiently nice - and $Vect$ is sufficiently nice.

But, just for my own peace of mind, let me explain that wrinkle I mentioned. A ring is not just a category with one object, it’s an $Ab$ -enriched category with one object. So, a bimodule of two rings $A$ and $B$ is not really a functor $F : A^{op} \times B \to Ab;$ it’s an $Ab$ -enriched functor $F : A^{op} \otimes B \to Ab .$

So, to make our general “bimodules” really subsume the usual concept of bimodule for rings, it’s best if we let them be, not functors $F : C^{op} \times D \to K$ where $C$ and $D$ are categories, but $K$ -enriched functors $F : C^{op} \otimes D \to K$ where $C$ and $D$ are $K$ -enriched categories!

Then everything in sight is $K$ -enriched, which makes life simpler in the end - we’re not hopping around between different contexts.

But, if your $C$ and $D$ are just ordinary categories, fear not! You can make them $K$ -enriched using any monoidal functor from $Set$ to $K$ .

For example, if $K = Vect$ , you can use the “free vector space on a set” functor. This amounts to taking your $C$ and $D$ and throwing in new morphisms that are linear combinations of the existing ones. We get some new $Vect$ -enriched categories $\tilde{C}$ and $\tilde{D}$ . Then any plain old functor $F : C^{op} \times D \to Vect$ becomes a $Vect$ -enriched functor $\tilde{F} : {\tilde{C}}^{op} \otimes \tilde{D} \to Vect$ so we’re working in a completely enriched world.

If you think what this means when $C$ and $D$ are groups, you’ll see that $\tilde{C}$ and $\tilde{D}$ are their group algebras! A representation of a group is the same as a representation of its group algebra, so nothing is lost; we’re just more firmly planted in the world of linear algebra now.

So, in fact, what the Australians really do is generalize the bicategory of

rings
bimodules
bimodule homomorphisms

to the bicategory of

$K$ -enriched categories
$K$ -enriched profunctors
$K$ -enriched natural transformations between $K$ -enriched profunctors

This is what they use whenever wimps like ourselves might be tempted to use

categories
functors
natural transformations

I’m exaggerating somewhat, but this is roughly their attitude. Remember, this is the country where Crocodile Dundee came from.

Posted by: John Baez on November 13, 2006 9:52 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

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Am I alone in wishing that the Australians would spend time exploiting this powerful apparatus to work out particular cases? Perhaps they are. If you read Lawvere’s article, mentioned here, you’ll see what can be achieved by ‘relentless pursuit’ even with a simple enriching $V$ such as the extended reals (p. 13 onwards). This makes clear that it’s not simply a case of automatically applying a general theory.

Posted by: David Corfield on November 14, 2006 8:58 AM | Permalink | Reply to this

Re: Klein 2-Geometry VII

David wrote:

Am I alone in wishing that the Australians would spend time exploiting this powerful apparatus to work out particular cases?

I’d rather let the Australians do what they do best, leaving the rest to us.

When one hears of “Australian category theorists”, one imagines massed ranks of powerful mathematicians. That indeed is the image the phrase is supposed to evoke. But, you can count them on two hands - and most of them are Ross Street.

When Street retires, I believe there will be no full professor at a research university in Australia who does category theory full time. Max Kelly is retired. I believe Steve Lack teaches at a school without mathematics grad students. Michael Johnson spends most of his energy on computer science. Brian Day has a loose affiliation with Macquarie, but not a paying job. Wesley Phoa is in finance; Dominic Verity was out of academia until recently - luckily he’s back in, and quite energetic. Alexei Davydov and the redoubtable Michael Batanin are research fellows. Do you know many more?

When Street retires, will anyone have sufficient clout to hire more people in this area? I hope so.

Luckily there are expatriate Australian category theorists like John Power and Mark Weber, honorary Australian category theorists like Martin Hyland - and would-be Australian category theorists like the people who attended Steve Lack’s talks at Chicago this spring, including Eugenia Cheng, Nick Gurski, Mike Shulman and me.

Posted by: John Baez on November 23, 2006 9:33 AM | Permalink | Reply to this

Re: Klein 2-Geometry VII

In that case.

It’s just that the wheels of n-category theory turn rather frustratingly slowly. Now we have you down on record saying that mathematical objects can be “intrinsically interesting”, you presumably also believe that the man hours devoted globally to mathematical research are not distributed fairly according to this intrinsic interest. A few more concrete successes might help redress the balance.

Posted by: David Corfield on November 23, 2006 3:51 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

David Corfield:

… the wheels of n-category theory turn rather frustratingly slowly.

You have a curiously impatient attitude towards mathematics.

Maybe I was like that too before I had lots of classes, half-written papers and books, grad students, grant proposals, committee work, letters of recommendation to write, conferences to organize, talks to plan for, travel to arrange - and blogging, shopping, socializing, gardening, eating, sleeping, etcetera to attend to. As a lonely youth, I had endless hours to spend on math. But now, getting to the point of doing math seems like pushing a large boulder encrusted with moss. Having ideas is quick. Proving theorems take a bit longer. But it’s all the other stuff that eats up vast amounts of time.

So, I’m learning to be patient.

And so, when you imagine “the Australian category theorists”, you should imagine Ross Street endlessly writing grant proposals and reports like this, this and this. That’s the reality.

Luckily, it’s not as if there’s really any rush to get everything figured out! I actually enjoy subjects when they’re mysterious more than when the solutions are all worked out. When I’m done unravelling a mystery, the fun is over and it’s time to move on.

… you presumably also believe that the man hours devoted globally to mathematical research are not distributed fairly according to this intrinsic interest.

No, nothing is perfect. In fact, most of the man hours supposedly devoted to mathematical research are actually spent doing chores of various sorts that prevent one from actually thinking. But that’s life!

Posted by: John Baez on November 23, 2006 8:45 PM | Permalink | Reply to this

Re: Klein 2-Geometry VII

You have a curiously impatient attitude towards mathematics.

Perhaps it’s just projected from my frustration with philosophy. As I mentioned in my post about my last conference:

Brendan Larvor, author of a very good book on Lakatos, opened his talk with “Here we are nearly 45 years after Lakatos’s Proofs and Refutations, still looking for the new epistemology…”. This struck a chord as Chapter 7 of my book opens “Nearly forty years have passed since Imre Lakatos published his paper Proofs and Refutations…”, itself updated from its earlier appearance as a journal article in 1997. We imagined ourselves 15 years hence, “Nearly 60 years have passed…”.

As you say,

Luckily, it’s not as if there’s really any rush to get everything figured out! I actually enjoy subjects when they’re mysterious more than when the solutions are all worked out. When I’m done unravelling a mystery, the fun is over and it’s time to move on.

A similar sentiment to Weil’s on working out an analogy:

Gone is the analogy: gone are the two theories, their conflicts and their delicious reciprocal reflections, their furtive caresses, their inexplicable quarrels; alas, all is just one theory, whose majestic beauty can no longer excite us. Nothing is more fecund than these slightly adulterous relationships; nothing gives greater pleasure to the connoisseur, whether he participates in it, or even if he is an historian contemplating it retrospectively, accompanied, nevertheless, by a touch of melancholy. The pleasure comes from the illusion and the far from clear meaning; once the illusion is dissipated, and knowledge obtained, one becomes indifferent at the same time

A host of balances to be struck then, including that between generating enough interest to bring in sufficiently many young people without creating a gold rush.

Posted by: David Corfield on November 24, 2006 10:10 AM | Permalink | Reply to this

Re: Klein 2-Geometry VII

When Street retires, I believe there will be no full professor at a research university in Australia who does category theory full time.

Don’t forget R.F.C. (“Bob”) Walters. These days, his and his students’ research interests tend strongly toward applications of category theory to CS, and he’s done quite a significant amount of work in pure category theory over the decades.

Posted by: Todd Trimble on November 24, 2006 2:26 AM | Permalink | Reply to this

Re: Klein 2-Geometry VII

Whoops – Walters is no longer listed as faculty at the University of Sydney; his home is now in Italy (I knew he’s had long-standing connections with the Italian categorical community). So AFAIK you’re right, John – Street is the only categorist who is a full professor at an Australian research university. Amazing.

Posted by: Todd Trimble on November 24, 2006 4:43 PM | Permalink | Reply to this

n-Quotients

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Turning my eyes regretfully down from the gods on the heights of Olympus, I shall continue ploughing my lonely furrow at its base.

I have been thinking about quotients recently. I think I understand some things better now, so I shall put down my observations in case they are helpful to anybody else.

Weak, firm and strict quotients

We start with a groupoid. We want to think of this as a ‘weak quotient’. This usually means the result of acting on a set with a group. For each member of the group that sends an element $a$ of the set to an element $b$ , we add one morphism $a \overset{f}{\to} b .$ However, if, instead of considering an action (functor) from a group to Set, we consider more generally a functor from a groupoid to Set, sending all objects in the groupoid to the same set in Set, then we can think of any groupoid we like as being a weak quotient.

On the other hand, we have also the “strict” quotient, which is the set that we get by setting all isomorphic objects in the groupoid equal to one another and throwing away the morphisms (except identities).

I’d like to insert an intermediate level quotient. In this, we have neither a set of morphisms between two objects (as with the weak quotient), nor a “tautology” of morphisms (i.e. only the necessary identity morphisms) as with the strict quotient, but a truth value of morphisms. That is to say, if the weak quotient has at least one morphism between two objects, the intermediate quotient has exactly one. This is nothing other than a set together with an equivalence relation on it. This is nice, since an equivalence relation is traditionally seen as the intermediate step between a group action and a partition considered as a set.

What we are doing here is taking the obvious functor from the weak quotient to the strict quotient and factorising it into a full bit and a faithful bit.

Instead of this, we can try doing the steps in a different order. Instead of first reducing a set of morphisms to just one (per pair of objects), and then identifying objects, we can instead try identifying the objects first. To do this, we need to pick a particular isomorphism between each pair of objects, and “contract it down to the identity”, dragging the two objects together into one, and forcing the other isomorphisms and automorphisms to merge in a particular way. This gives us a set of groups. We can, if we wish, then force these down into a mere set of elements by identifying all automorphisms of a given object.

The fact that we can do this, merging at the 0-morphism level while failing to do so at the 1-morphism level, makes me wonder if perhaps we should have had a preliminary stage in which we add one 2-morphism from each 1-morphism to each other 1-morphism between the same objects, and then identified 1-morphisms by contraction of the 2

The n-Category Café

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November 1, 2006